Abstract: I will discuss the rational cohomology of SL_nR, Sp_{2n}R, and their principal congruence subgroups for R a number ring. Borel--Serre showed that these groups satisfy a (co)homological duality that lets us study their cohomology groups via certain representations called the `Steinberg modules’, which have a beautiful combinatorial description in terms of Tits buildings.
I will describe a conjecture of Lee--Szczarba on the top cohomology of principal congruence subgroups of SL_nZ, and its resolution due to Miller--Patzt--Putman. I will then discuss forthcoming work on generalizations of this to other Euclidean rings, and also to symplectic groups.